Single-step quantitative susceptibility mapping (QSM) algorithms simplify the processing pipeline and promise to be more robust against background fields than traditional two-step methods but they often underestimate tissue susceptibilities. Here, we propose a highly efficient gradient descent Tikhonov-regularized proximal solver and a highly accurate ADMM TV-regularized proximal solver to improve the accuracy of two Laplacian-based single-step methods. Our solvers outperformed current single-step methods and showed in-vivo performance very similar to traditional two-step methods. This will simplify QSM processing pipelines, allowing further automation in future, although more research is needed to improve robustness against noise and boundary-conditions-related artifacts.
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